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Chapter 2 Mercator, Wright and Mapmaking

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Chapter 2 Mercator, Wright and Mapmaking © Paul J. Lewi, 2006 Version of February 11, 2006 Speaking of Graphics Chapter 2 Mercator, Wright and Mapmaking - 1 - 2.1 Geographic maps as bivariate graphics In the chapter on quipu we have described the univariate nature of the accounting system of centralized Inca government. Univariate means that a collection of objects (or subjects) is described from a single point of view: so much of this, so much of that, and so on. Because of their lack of written and printed communication, the Incas were not confronted with the two-dimensional space of tablets, parchment and paper. The latter offer the possibility for a bivariate graphic, in which some part of the world is viewed simultaneously from two points of view. It appears that the bivariate graphic display has been independently developed in Europe and in Asia. The most obvious one is the geographic map in which points on the earth are represented by two of their geographical properties, namely longitude and latitude as measured on the globe. Today, we may regard the problem of mapping the earth as rather trivial, and we easily forget the long and arduous way which has led to modern cartography. We often disregard the inherent problems of mapping a three-dimensional object such as a sphere onto a plane. We also tend to blind ourselves to the inherent bias and subjectivity that is unavoidable in any visual display of our experience of a multidimensional reality. For this reason it seems worthwhile to investigate briefly the early history of mapmaking. The contributions of Gerard Mercator and Edward Wright will be discussed more extensively, because they discovered and applied the mathematics of projecting the globe into a plane. Geographic maps have led to the development of functional graphics and of statistical charts. Functional graphics describe a property of an object (or subject) as a theoretical function of another property, such as the change of velocity of an object with respect to time, the evolution of the composition of a chemical mixture as a function of temperature, and so on. Statistical charts relate two properties that - 2 - have been measured (or observed) in a set of objects (or subjects), such as mortality from heart disease and consumption of fat in various countries. This may answer questions about the relationship between mortality and fat consumption in the general population. The distinction between functional graphics and statistical charts is slight, however. Both require a degree of abstraction which leads to the representation of non-geometric properties (such as velocity, time, temperature, concentration, mortality, consumption, etc.) in a geometrical way. The transition from cartography to more general graphics and charts has also been long and tortuous. Eventually it led to the great synthesis between algebra and geometry, known as coordinate geometry, which was proposed by René Descartes around 1637, and which is discussed in a following chapter. In the meantime we sketch summarily the main events and concepts that have paved the way for the bivariate diagram. - 3 - 2.2 Ancient Mapmaking The starting point of our brief journey to the world of mapmaking lies in the 6th century BC in Greece, when Pythagoras and the Pythagorean School after him realized that the earth was a spherical object, although they had no idea of its size. In the 4th century BC, Aristotle defined six climatic parallels on the terrestrial globe (‘climata’) and introduced the notion of latitude as a property of places on earth. In early days, degrees of latitude were determined from the shadow cast by the sun at its highest point in the sky from a ‘gnomon’, basically a stick placed vertically in the soil. Later on, in the 3rd century BC, the Greek astronomer and geographer Eratosthenes, who worked at the library of Alexandria in Egypt, defined longitudes through well-known places such as Alexandria and Rhodes. In that time, longitudes were estimated from travel reports of voyagers and soldiers, especially from the campaigns of Alexander the Great. Eratosthenes also arrived at a remarkably accurate estimate of the perimeter of the earth. The latter was deduced from the difference of the heights of the sun at the summer solstititium in Alexandria and in Syene (today called Aswan). These two cities are situated approximately on the same meridian and their distance was estimated at 800 km. The difference of the heights of the sun at noon was determined as 1/50th of a full circle. Hence, the perimeter of the earth turned out to be 800 x 50 or 40,000 km. Although Eratosthenes' observations were rather crude, he had the good fortune that his errors cancelled out [1]. Once the length of the equator was known, one could convert distances along parallels into degrees of longitude by means of trigonometry. In the course of the first century BC, the Greek stoic philosopher Posidonius repeated Eratosthenes' observations, but using this time the declination of the bright star Canopus when it was just above the horizon. Unfortunately, this time, measurement errors did not cancel out and Posidonius' estimate of the length - 4 - of the equator turned out to be 29,000 km, some 11,000 km shorter than that of Eratosthenes [2]. Subsequent scholars, among which Marinus of Thyrus, adopted the new estimate by Posidonius and, as we will see, the error survived until 1492 when Columbus set sail on a presumably shorter western route to China and India. Note that the use of a rectangular grid defined by degrees of longitudes and degrees of latitudes was already practiced by geographers in Posidonius' time. Figure 2.1. Woodcut reproduction of Ptolemaeus’s map of the world by Nicolaus Germanus with curved meridians and parallels, from a German printed edition of the Cosmographia, which appeared in Ulm around 1482. The characteristic features of the Ptolemaic maps are the landlocked Indian Ocean, the absence of the Indian peninsula, the large island Taprobana, probably representing Ceylon, and the elongated shape of the Mediterranean Sea and of Scotland [3]. The most famous of the early Greek geographers and astronomers was Claudius Ptolemaeus (Ptolemy), who worked at the library of Alexandria, probably from 125- 150 AD. Ptolemaeus is best known for his geocentric, more precisely geostatic, - 5 - vision of the universe, which he described in his Megale Synthesis (Great Synthesis) or Almagest (from the Arabic Al Madjisti). His vision became the established doctrine well until the 16th century, when it was gradually replaced by Copernicus' heliocentric model. The Cosmographia (also referred to as Geographia) of Ptolemaeus contained a catalog of 8,000 places, of which some 400 were provided with (gnomonic) latitudes and with longitudes that were computed from traveler accounts. It also contained 27 maps, including one world map and 26 local ones. Ptolemaeus projected his world upon a cone which touched the earth globe at the 63rd parallel. The latter ran through the legendary island of Thule, the most northern part of the then known world (Ultima Thule has been identified with the Shetland Islands, the west coast of Norway and Iceland). He placed the zero meridian through the Fortunate islands (now called Canaries). This way, Ptolemaeus defined a conical grid (or graticule) which corrected for the shortening of the parallels at greater latitudes. In a first version, the parallels were drawn as curved lines, while the meridians are represented as straight lines. In a later version, both the parallels and the meridians appear to be curved (Fig. 2.1). Ptolemaeus's Cosmographia is the earliest attempt to separate facts from fiction, to present a coherent view of the world and to deal with the two-dimensional constraint of the plane. The Cosmographia certainly can be regarded as a milestone in mapmaking and has prepared the way toward more general coordinate-based charts. It provided the theoretical framework from which modern maps have been derived. - 6 - Figure 2.2. Coordinate system used by the Romans for cadastral mapping, called centuriation. Plots of land are defined by their position with respect to the Decumanus Maximus (DM) and Kardo Maximus (KM). Adapted from A. Hodgkiss [3]. In the Roman area, which extended into the 5th century AD, the great tradition initiated by Ptolemaeus came to a halt in Western Europe. It was only to be revived, as we will see, during the 15th century Renaissance, when the Cosmographia was rediscovered in Byzantium and brought to Italy. The Romans devoted more of their attention to practical matters. Their requirements for traveling were largely covered by so-called itineraries, which bore no relation with the actual geographic positions and directions. The Romans, however, invented a - 7 - system of rectangular coordinates for the purpose of cadastral mapping and surveying, which was called ‘centuriation’. (Centuria is the Latin term for something that can be divided into hundred parts.) Its purpose was to divide newly won territories in rectangular plots for the purpose of colonization. Surveyors (agrimensors) made this division relative to a horizontal orientation called ‘Kardo’ (which means pen or fulcrum) and a vertical direction which was referred to as ‘Decumanus’ (which means line of division) [3]. The Roman grid system is illustrated in Fig. 2.2. Each plot is identified by its horizontal and vertical coordinates (Kardines and Decumani). For example, the notation SDII and VKI means two positions at the left (S) of the Decumanus Maximus (D) and at the same time one (I) position above (V) the Kardo Maximus (K). In this case the coordinate system is used as a two-dimensional index for retrieving information on a map. It is perhaps one of the oldest examples of a bivariate display of information on a rectangular grid.
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